The discussion centers on the necessity of the second countable and Hausdorff conditions in the definition of manifolds. These properties ensure that a topological manifold is locally Euclidean and allows for the application of Urysohn's metrization theorem, which states that every Hausdorff second-countable regular space is metrizable. The importance of these conditions is underscored by their role in enabling partitions of unity and ensuring that manifolds can be embedded in Euclidean spaces. Pathological examples like the "line with two origins" and the "long line" illustrate the limitations of dropping these conditions.
PREREQUISITESMathematicians, particularly those specializing in topology and differential geometry, as well as students seeking to understand the foundational aspects of manifold theory.
My first thought was: We need a reasonable partition of unity in which the sums are indexed by the natural numbers. This would probably require even normal topologies. The two specific properties (second countability aka complete separability and Hausdorff) both occur in Urysohn's metrization theorem:littlemathquark said:TL;DR Summary: Why do we need second countable and Hausdorff conditions for manifold definition?
Why do we need second countable and Hausdorff conditions for manifold definition?
However, ...One of the first widely recognized metrization theorems was Urysohn's metrization theorem. This states that every Hausdorff second-countable regular space is metrizable.
... and here we are, back at the partition of unity.It follows that every such space is completely normal as well as paracompact.
Indeed. I was so used to reading Riemannian and even smooth, nice charts, metrics, and spheres that I almost forgot the topological aspect. The two concepts of Urysohn are somehow the minimum we want to have.dextercioby said:"Manifold" is improper usage. One has "topological manifolds" and a very important subset of all of these, the "complex or real differential manifolds".
That is a strange question! Why do you need the other conditions in the definition of a manifold? You can study whatever objects you want, why is a question only you can answer. Some people do look at non Hausdorff manifolds. https://en.wikipedia.org/wiki/Non-Hausdorff_manifoldlittlemathquark said:TL;DR Summary: Why do we need second countable and Hausdorff conditions for manifold definition?
Why do we need second countable and Hausdorff conditions for manifold definition?
Yes, but these are all local properties. It gives you information about open sets at a certain location. The atlas patches all these local information. The result is a book (an atlas) of maps (the homeomorphisms). But this is literally a patchwork. Urysohn's results (partition of unity and metrization) are global properties for the entire manifold.littlemathquark said:In the definition of a topological manifold, the conditions that the topological space X be Hausdorff and second countable are not always included. Is this because these two properties can be transferred via a homeomorphism from R^n, since they are topological properties?
I am the wrong person to answer that. I believe that particularly topology is a huge collection of pathological examples. It can trick someone's intuition in so many cases! E.g. I cannot really imagine how we can color a square with a pen that has no width. As I mentioned above, the "nice" manifolds are Riemannian, and smooth, and the only question is whether they can be embedded in ##\mathbb{R}^{n+1}## or ##\mathbb{R}^{2n}.##littlemathquark said:However, I believe these are pathological examples.
We only need partitions of unity for smooth manifolds so we can develop a reasonable coordinate-free theory of integration. For topological manifolds, we don't have such a theory so we don't need partitions of unity. Nevertheless the condition here of second countability is enough to prove that topological manifolds are paracompact and this in turn is enough to prove we have partitions of unity. This is because topological manifolds are always locally compact and local compactness plus second countability is enough to prove it must be sigma compact, and then local compactness and sigma compact is enough to prove paracompact and hence partitions of unity.fresh_42 said:My first thought was: We need a reasonable partition of unity in which the sums are indexed by the natural numbers. This would probably require even normal topologies. The two specific properties (second countability aka complete separability and Hausdorff) both occur in Urysohn's metrization theorem:
However, ...
... and here we are, back at the partition of unity.
Both are important tools we do not want to lose.
Thank you. I think this is also true: Every manifold is homeomorphic to a suitable subspace of a Euclidean space, and an uncountable discrete set cannot be embedded into any Euclidean space. Therefore, the second countability condition in the definition of a manifold is an indispensable condition if we aim to embed manifolds into Euclidean space. Euclidean space and every subspace of it are Hausdorff and second countable. It can be shown that a differentiable manifold ##M## of dimension ##n## can be embedded into ##R^{2n+1}## by fundamentally using the fact that it is Hausdorff and second countable.Mozibur Rahman Ullah said:The most general definition of a manifold is that of a topological manifold. Now these are defined as Hausdorff and second countable topological spaces that are locally Euclidean.
Now if we drop second countability and Hausdorff condition from the definition of a topological manifold we get what is called a locally Euclidean space. Why don't we use this as a definition of a manifold? For two reasons.
First, given that Euclidean spaces are Hausdorff, we can hope that the Hausdorff property of the Euclidean space is transferred to a locally Euclidean space. But this isn't the case by the counterexample of the line with two origins. This is locally Euclidean but is not Hausdorff. Thus motivates introducing the Hausdorff condition.
Another pathological example is the long line. We first build the long ray by taking the first uncountable ordinal ##\omega_1## and inserting the interval ##(0,1]## in between every two numbers in this ordinal. We then glue two long rays together and this yields the long line. This is again locally Euclidean but we don't want to count this a manifold as it is infinitely longer than the real line. To discount this we assume the second axiom - that it is second countable. This means that its topology has a countable base.
Thus a topological manifold is a second countable and Hausdorff locally Euclidean space.
I hope this helps.
If you have paracompactness, this allows you to create paritions of unity subordinate to a cover that allow you to put together coherently all these local objects.littlemathquark said:In the definition of a topological manifold, the conditions that the topological space X be Hausdorff and second countable are not always included. Is this because these two properties can be transferred via a homeomorphism from R^n, since they are topological properties?
An uncountable discrete set in Euclidean n-space will necessarily have a limit point there, by Weirstrass' Theorem: Every bounded infinite subset of the plane contains a limit point. You can then,,e.g., cover the plane by balls B(0,n): n=1,2,... . By cardinality, one of these must contain infinitely many points.littlemathquark said:Thank you. I think this is also true: Every manifold is homeomorphic to a suitable subspace of a Euclidean space, and an uncountable discrete set cannot be embedded into any Euclidean space. Therefore, the second countability condition in the definition of a manifold is an indispensable condition if we aim to embed manifolds into Euclidean space. Euclidean space and every subspace of it are Hausdorff and second countable. It can be shown that a differentiable manifold ##M## of dimension ##n## can be embedded into ##R^{2n+1}## by fundamentally using the fact that it is Hausdorff and second countable.